Fundamentals of Hypersonic Vehicle Design Part II: ThermoStructural Analysis

1
Fundamentals of Hypersonic Vehicle DesignPart
II Thermo-Structural Analysis

• Frederick Ferguson
• North Carolina AT State University
• September 13, 2007

Sect 3.1.2
2
3.1.2 Thermo-Structural Analysis
3.1.2.1 Thermo-Structural Perspective of Vehicle
Design 3.1.2.2 Review of Strength of
Materials 3.1.2.3 The Mechanics of 1D Linear
Elastic Materials 3.1.2.3.1 Illustrative
Examples of Linear Elastic Materials 3.1.2.4 1D
Linear Thermo-Elastic Problems with
Applications 3.1.2.5 Formulation of the
Thermo-Elastic Problem 3.1.2.5.1 Illustrative
Examples of the Thermo-Elastic Problem References
Sect 3.1.2
3
3.1.2 Thermo-Structural Analysis
3.1.2.1 Thermo-Structural Perspective of Vehicle
Design 3.1.2.2 Review of Strength of
Materials 3.1.2.3 The Mechanics of 1D Linear
Elastic Materials 3.1.2.3.1 Illustrative
Examples of Linear Elastic Materials 3.1.2.4 1D
Linear Thermo-Elastic Problems with
Applications 3.1.2.5 Formulation of the
Thermo-Elastic Problem 3.1.2.5.1 Illustrative
Examples of the Thermo-Elastic Problem
Sect 3.1.2
4
Structural Design Aspects of Hypersonic Vehicle

• Structural Strength
• The capacity of the individual elements, which
  together make up the hypersonic structural
  system, to withstand the load (static, dynamic
  transient) that is applied to it.
• Structural Stability
• The capability of a structural system to transmit
  the aero thermal loads safely to the neighboring
  members and eventually to the mainframe.

Sect 3.1.2
5
Structural Design Objectives

• Safely Transfer all aerothermodynamics forces to
  the aircraft mainframe (perpetually maintaining
  system strength stability)

Contrasting Structural Design Approaches
Sect 3.1.2
6
Fundamentals Structural Loads and Structural
Responses
Stress Tensor Internal Stresses due to Applied
Loads
External Force ? Internal Stresses
Displacements ? Strains
Strain Tensor The development of Strains due to
Internal Stresses
Constitutive Relations
Solution Approach 1D Concepts, then
generalized to 3D
Sect 3.1.2
7
3.1.2 Thermo-Structural Analysis
3.1.2.1 Thermo-Structural Perspective of Vehicle
Design 3.1.2.2 Review of Strength of
Materials 3.1.2.3 The Mechanics of 1D Linear
Elastic Materials 3.1.2.3.1 Illustrative
Examples of Linear Elastic Materials 3.1.2.4 1D
Linear Thermo-Elastic Problems with
Applications 3.1.2.5 Formulation of the
Thermo-Elastic Problem 3.1.2.5.1 Illustrative
Examples of the Thermo-Elastic Problem
Sect 3.1.2
8
Strength of Materials The Basics

• When a material is subjected to a load (force),
  it is distorted or deformed.
• If the load is small, the distortion will
  probably disappear when the load is removed. The
  intensity, or degree, of distortion is known as
  strain.
• When a load is applied to the material, the
  structure itself is strained, as it is being
  compressed, warped and/or extended in the
  process.
• If the distortion disappears and the metal
  returns to its original dimensions upon removal
  of the load, the strain is called elastic strain.
  
• If the distortion disappears and the material
  remains distorted, the strain type is called
  plastic strain.

Sect 3.1.2
9
Properties of Materials
Properties of Materials An elastic modulus, or
modulus of elasticity, E, is the mathematical
description of an object or substance's tendency
to be deformed elastically (i.e. non-permanently)
when a force is applied to it. The elastic
modulus of an object is defined as the slope of
its stress-strain curve in the elastic
deformation region

• where E is the elastic modulus stress is the
  force causing the deformation divided by the area
  to which the force is applied and strain is the
  ratio of the change caused by the stress to the
  original state of the object. If stress is
  measured in Pascals, and since strain is a
  unitless ratio, then the units of E are pascals
  as well.
• Specifying how stress and strain are to be
  measured, including directions, allows for many
  types of elastic moduli to be defined. The three
  primary ones are
• Young's modulus (E) describes tensile elasticity,
  or the tendency of an object to deform along an
  axis when opposing forces are applied along that
  axis it is defined as the ratio of tensile
  stress to tensile strain. It is often referred to
  simply as the elastic modulus.
• The shear modulus or modulus of rigidity (G)
  describes an object's tendency to shear (the
  deformation of shape at constant volume) when
  acted upon by opposing forces it is defined as
  shear stress over shear strain.
• The bulk modulus (K) describes volumetric
  elasticity, or the tendency of an object's volume
  to deform when under pressure it is defined as
  volumetric stress over volumetric strain, and is
  the inverse of compressibility. The bulk modulus
  is an extension of Young's modulus to three
  dimensions.

Sect 3.1.2
10
The Nature of Material Stress
Stresses can be generally classified in one of
six categories Residual Stress Residual
stresses are due to the manufacturing processes
that leave stresses in a material. Welding leaves
residual stresses in the metals
welded.  Structural Stress Structural stresses
are stresses produced in structural members
because of the weights they support. The weights
provide the loadings. These stresses are found in
building foundations and frameworks, as well as
in machinery parts. Pressure Stress Pressure
stresses are stresses induced in vessels
containing pressurized materials. The loading is
provided by the same force producing the
pressure. Flow Stress Flow stresses occur when a
mass of flowing fluid induces a dynamic pressure
on a conduit wall. The force of the fluid
striking the wall acts as the load. This type of
stress may be applied in an unsteady fashion when
flow rates fluctuate. Thermal Stress Thermal
stresses exist whenever temperature gradients are
present in a material. Different temperatures
produce different expansions and subject
materials to internal stress. This type of stress
is particularly noticeable in mechanisms
operating at high temperatures that are cooled by
a cold fluid.  Fatigue Stress Fatigue stresses
are due to cyclic application of a stress. The
stresses could be due to vibration or thermal
cycling. 
Sect 3.1.2
11
Types of Internal Stress
The Stress Intensity within the material is
expressed as one of three basic types of internal
force. They are known as either tensile,
compressive, or shear force. Mathematically,
there are only two types of internal load because
tensile and compressive stress may be regarded as
the positive and negative versions of the same
type of normal loading. However, in mechanical
design, the response of components to the two
conditions can be so different that it is better,
and safer, to regard them as separate types. The
plane of a tensile or compressive stress lies
perpendicular to the axis of operation of the
force from which it originates. The plane of a
shear stress lies in the plane of the force
system from which it originates. It is essential
to keep these differences quite clear both in
mind and mode of expression.
Sect 3.1.2
12
Types of Internal Stress
Tensile Stress Tensile stress is that type of
stress that cause two sections of material on
either side of a stress plane to pull apart or
elongate. Compressive Stress Compressive stress
is the reverse of tensile stress. Adjacent parts
of the material tend to press against each other
through a typical stress plane. Shear
Stress Shear stress exists when two parts of a
material tend to slide across each other in any
typical plane of shear upon application of force
parallel to that plane.
Sect 3.1.2
13
Stresses in Materials
Assessment of mechanical properties is made by
addressing the three basic stress types
compressive, tensile and shear. Normal
Stresses Because tensile and compressive loads
produce stresses that act across a plane, in a
direction perpendicular (normal) to the plane,
tensile and compressive stresses are called
normal stresses. Compressibility The ability of
a material to react to compressive stress (or
pressure) is called compressibility. For example,
metals and liquids are incompressible, but gases
and vapors are compressible. Two types of
stress can be present simultaneously in one
plane, provided that one of the stresses is shear
stress. Under certain conditions, different
basic stress type combinations may be
simultaneously present in the material. An
example would be a reactor vessel during
operation. The wall has tensile stress at various
locations due to the temperature and pressure of
the fluid acting on the wall. Compressive
stress is applied from the outside at other
locations on the wall due to outside pressure,
temperature, and constriction of the supports
associated with the vessel. In this situation,
the tensile and compressive stresses are
considered principal stresses.
Sect 3.1.2
14
Mechanics of Materials
Whenever a stress (no matter how small) is
applied to an elastic material/structure, a
proportional dimensional change or distortion
takes place, Hookes Law.  Such a proportional
dimensional change is called strain and is
measured as the total elongation per unit length
of material due to some applied stress, and
evaluated as follows
where e strain (in./in.) d total
elongation (in.) L original length (in.)
Types of Strain Strain may take two forms
elastic strain and plastic deformation.  Elastic
Strain Elastic strain is a transitory dimensional
change that exists only while the initiating
stress is applied and disappears immediately upon
removal of the stress. Elastic strain is also
called elastic deformation. The applied stresses
cause the atoms in a crystal to move from their
equilibrium position. All the atoms are displaced
the same amount and still maintain their relative
geometry. When the stresses are removed, all the
atoms return to their original positions and no
permanent deformation occurs. Plastic
Deformation Plastic deformation (or plastic
strain) is a dimensional change that does not
disappear when the initiating stress is removed.
It is usually accompanied by some elastic strain.
Sect 3.1.2
15
Structural Responses to External Loads
In general Structural responses to external loads
are manifested in a combination of the following
form Displacement (translation motion, rotation,
and distortion), and internal stresses
Consider a rod with initial length L which is
stretched to a length L'. The strain measure e, a
dimensionless ratio, is defined as the ratio of
elongation with respect to the original length,
Material Properties
The Elastic Modulus, E, and the Shear Modulus, S,
are material properties that resist shear and
shear stresses.
Sect 3.1.2
16
Elasticity and Plasticity
The phenomenon of elastic strain and plastic
deformation in a material are called elasticity
and plasticity, respectively. At room
temperature, most materials have some elasticity,
which manifests itself as soon as the slightest
stress is applied. Usually, they also possess
some plasticity, but this may not become apparent
until the stress has been raised appreciably.
The magnitude of plastic strain, when it does
appear, is likely to be much greater than that of
the elastic strain for a given stress increment.
Metals are likely to exhibit less elasticity and
more plasticity at elevated temperatures. The
state of stress just before plastic strain begins
to appear is known as the proportional limit, or
elastic limit, and is defined by the stress level
and the corresponding value of elastic strain.
The proportional limit is expressed in force
per unit length. For load intensities beyond the
proportional limit, the deformation consists of
both elastic and plastic strains. Strain measures
the proportional dimensional change with no load
applied. Such values of strain are easily
determined and only cease to be sufficiently
accurate when plastic strain becomes dominant.
Sect 3.1.2
17
3.1.2 Thermo-Structural Analysis
3.1.2.1 Thermo-Structural Perspective of Vehicle
Design 3.1.2.2 Review of Strength of
Materials 3.1.2.3 The Mechanics of 1D Linear
Elastic Materials 3.1.2.3.1 Illustrative
Examples of Linear Elastic Materials 3.1.2.4 1D
Linear Thermo-Elastic Problems with
Applications 3.1.2.5 Formulation of the
Thermo-Elastic Problem 3.1.2.5.1 Illustrative
Examples of the Thermo-Elastic Problem
Sect 3.1.2
18
Material Response Hookes Law
s Ee
In elastic material the stress is directly
proportional to the strain The Elastic Modulus
or Young's Modulus, E, is equal to the ratio of
Stress to Strain, over the elastic limit The
elastic modulus is a fundamental property of
material to directly resist that actions of
stresses
19
Shear Strain
Axial stress results in an axial strain, and so
does shear stress produce a shear strain. Both
Axial Strain and Shear Strain are shown. The
shear stress produces a displacement of the rod
as indicated in the Diagram. The edge of the rod
is displaced a horizontal distance from its
initial position. This displacement divided by
the length of the rod L is equal to the Shear
Strain. Examining the small triangle made by
delta L and the side of the rod, we see that the
Shear Strain, dL/L, is also equal to the tangent
of the angle gamma. Since the amount of
displacement is quite small, the tangent of the
angle is approximately equal to the angle g. As
with Axial Stress and Strain, Shear Stress and
Strain are proportional in the elastic region of
the material. This relationship may be expressed
as G Shear Stress/Shear Strain, where G is a
property of the material and is called the
Modulus of Rigidity (or at times, the Shear
Modulus). If a graph is made of Shear Stress
versus Shear Strain, it will normally exhibit the
same characteristics as the graph of Axial Stress
versus Axial Strain.
Sect 3.1.2
20
Mechanics of Materials
Sect 3.1.2
21
Mechanics of Materials
Thermal Stress (Obtained from Hookes Law
Thermal Strain)
Sect 3.1.2
22
Thermal expansion in Materials
Thermal expansion Coefficient
In physics, thermal expansion is the tendency of
matter to increase in volume. For liquids and
solids the amount of expansion will normally vary
depending on the material's coefficient of
thermal expansion. When objects expand tensile
forces are created. When objects contract
compressive forces are created. To accurately
calculate thermal expansion of a substance a more
advanced Equation of state must be used. For
solid materials with a significant length, like
beams or bars, an estimate of the amount of
thermal expansion can be described by the ratio
of strain
is the initial length before the change of
temperature and
the final length recorded after the change of
temperature.
For most solids, thermal expansion relates
directly with temperature
Thus, the change in either the strain or
temperature can be estimated by
where
is the coefficient of thermal expansion in
inverse kelvins.
is the difference of the temperature between the
two recorded strains, measured in Celsius or
Kelvin.
Sect 3.1.2
23
Shear in cantilever beam

• Shear stress is a stress state where the stress
  is parallel or tangential to a face of the
  material, as opposed to normal stress when the
  stress is perpendicular to the face. The variable
  used to denote shear stress is t (tau). The
  formula for shear stress in a beam is
•            
• V shear force at that location
• Q first moment of area
• t thickness in the material perpendicular to
  the shear
• I second moment of area of the cross section.
• The magnitudes and directions of V and M may be
  obtained form the equations of
• equilibrium Fy 0 and M 0, where O is any axis
  perpendicular to plane xy (the reaction R must be
  evaluated first from the free body of the entire
  beam). For the present the shearing stresses will
  be ignored while the normal stresses are studied.

Sect 3.1.2
24
Bending Stress The Flexure Formula in Beams
Consider the bending stress which develops in a
loaded beam. In Diagram 1 we have shown a simply
supported beam loaded at the center. The beam
responds by deflecting (or bending) under the
load as shown in Diagram 2.
Diagram 1
Diagram 2
Flexure Formula Stresses calculated from the
flexure formula are called bending stresses or
flexural stresses.
Sect 3.1.2
25
ELASTIC BENDING THEORY
Assumptions that were made in order to develop
the Elastic Theory of Bending. The beam has a
constant, prismatic cross-section and is
constructed of a flexible, homogenous material
that has the same Modulus of Elasticity in both
tension and compression (shortens or elongates
equally for same stress). The material is
linearly elastic the relationship between the
stress and strain is directly proportional. The
beam material is not stressed past its
proportional (elastic) limit. A plane section
within the beam before bending remains a plane
after bending. The neutral plane of a beam is a
plane whose length is unchanged by the beam's
deformation. This plane passes through the
centroid of the cross-section.
Sect 3.1.2
26
The Nature of Bending Stresses
Strain at a distance y above the neutral axis
Bending Stress (Hooks law)
Resultant Bending Moment
where
Sect 3.1.2
27
3.1.2 Thermo-Structural Analysis
3.1.2.1 Thermo-Structural Perspective of Vehicle
Design 3.1.2.2 Review of Strength of
Materials 3.1.2.3 The Mechanics of 1D Linear
Elastic Materials 3.1.2.3.1 Illustrative
Examples of Linear Elastic Materials 3.1.2.4 1D
Linear Thermo-Elastic Problems with
Applications 3.1.2.5 Formulation of the
Thermo-Elastic Problem 3.1.2.5.1 Illustrative
Examples of the Thermo-Elastic Problem
Sect 3.1.2
28
Structural Analysis of A Weight Optimized Wing
29
Structural Analysis of the Wing
Shear Stress in a Given Cross Section
Schematic of Pressure Variation along the z-axis
Schematic of Pressure Variation along the x-axis
Sect 3.1.2
30
Evaluation of the Required Moments of Inertia
CENTROIDAL COORDINATES (xG,yG)
MOMENT OF INERTIA
Sect 3.1.2
31
EVALUATE THE SHEAR CENTER
BENDING STRESS AT A GIVEN Z-LOCATION
WHERE,
Sect 3.1.2
32
Evaluation of the Local Shear Force and Moments
at Wing C-Section
Z_wing Tip
Z
Z_wing Tip
z_Wing_Root
Z
z_Wing_Root
Sect 3.1.2
33
EVALUATION OF MAXIMUM SHEAR STRESS AT A GIVEN
C-SEC
BENDING STRESS AT A GIVEN Z-LOCATION
A
C
D
B
Sect 3.1.2
34
Evaluation of the Maximum Shear Stress in the
Wing Ribs
Sect 3.1.2
35
Wing Structures Analyzed Optimized for Minimum
Weight
Sect 3.1.2
Ref F. Ferguson/S. Akwaboa
36
3.1.2 Thermo-Structural Analysis
3.1.2.1 Thermo-Structural Perspective of Vehicle
Design 3.1.2.2 Review of Strength of
Materials 3.1.2.3 The Mechanics of 1D Linear
Elastic Materials 3.1.2.3.1 Illustrative
Examples of Linear Elastic Materials 3.1.2.4 1D
Linear Thermo-Elastic Problems with
Applications 3.1.2.5 Formulation of the
Thermo-Elastic Problem 3.1.2.5.1 Illustrative
Examples of the Thermo-Elastic Problem
Sect 3.1.2
37
Influence of External Forces And Temperature
The strain in an object due to simultaneous
action of the external force and the temperature
change are described as follows
Using the generalized Hookes Law and the physics
of thermal expansion, the engineering expressions
for the normal and shear strains can be expressed
as follows
E Youngs Modulus G Shear Modulus DT
Temperature Change
Sect 3.1.2
38
Example Thermal Stress in Clamped Bar
Sect 3.1.2
39
Sect 3.1.2
40
Sect 3.1.2
41
Clamped Composite Bar with Uniform Temperature
Change
Sect 3.1.2
42
Sect 3.1.2
43
Sect 3.1.2
44
Sect 3.1.2
45
3.1.2 Thermo-Structural Analysis
3.1.2.1 Thermo-Structural Perspective of Vehicle
Design 3.1.2.2 Review of Strength of
Materials 3.1.2.3 The Mechanics of 1D Linear
Elastic Materials 3.1.2.3.1 Illustrative
Examples of Linear Elastic Materials 3.1.2.4 1D
Linear Thermo-Elastic Problems with
Applications 3.1.2.5 Formulation of the
Thermo-Elastic Problem 3.1.2.5.1 Illustrative
Examples of the Thermo-Elastic Problem
Sect 3.1.2
46
Stress And Traction
Stress is conventionally defined as a force
acting on some area
A traction is a force per unit area acting on a
specified surface
Traction, s, acting on the area defined by the
plane AA'-A'C'-C'C-CA can be resolved into a
normal component (sn) and parallel component (t).

• A Requirement for Continuum, Differential
  Tensor Analysis

Sect 3.1.2
47
The State of Stress Definition of Traction
The concept of stress originated from the study
of strength and failure of solids. The stress
field is the distribution of internal "tractions"
that balance a given set of external tractions
and body forces. First, we look at the external
traction T that represents the force per unit
area acting at a given location on the body's
surface. Traction T is a bound vector, which
means T cannot slide along its line of action or
translate to another location and keep the same
meaning. In other words, a traction vector cannot
be fully described unless both the force and the
surface where the force acts on has been
specified. Given both DF and Ds, the traction T
can be defined as
The internal traction within a solid, or stress,
can be defined in a similar manner. Suppose an
arbitrary slice is made across the solid shown in
the above figure, leading to the free body
diagram shown at right. Surface tractions would
appear on the exposed surface, similar in form to
the external tractions applied to the body's
exterior surface. The stress at point P can be
defined using the same equation as was used for
T. Stress therefore can be interpreted as
internal tractions that act on a defined internal
datum plane. One cannot measure the stress
without first specifying the datum plane.
Sect 3.1.2
48
Volume Surface Forces
Consider the Volume Force, P (Force/unit volume),
and its projections on the cubic element
Consider the Surface Force, P (Force/unit area),
applied to the center of each surface making up
the control volume
Sect 3.1.2
49
Surface-Forces Stress Tensor
Consider the Tetrahedron with surface SABC with
a normal, v, and with Surfaces, Sx, Sy and Sz,
with normals in the x, y and z directions, then
Projection of the Stresses on the surface SABC
with a normal, v
The Stresses tensor
Sect 3.1.2
50
The Equilibrium Equations
Consider the Tetrahedron with surface SABC with
a normal, v, and with Surfaces, Sx, Sy and Sz,
with normals in the x, y and z directions, then
if the surface is in equilibrium
The equilibrium equation in the x-direction
The equation of motion for a continuous medium in
the x-direction
Sect 3.1.2
51
Normal and Shear Strains
The engineering shear strain, gxy is a total
measure of shear strain in the x-y plane. In
constrast, the shear strain exy is the average of
the shear strain on the x face along the y
direction, and on the y face along the x
direction.
where
Sect 3.1.2
52
Strain-Displacement Relations
The infinitesimal strain-displacement
relationships can be summarized as,
where u is the displacement vector, x is
coordinate, and the two indices i and j can range
over the three coordinates 1, 2, 3 in three
dimensional space. Expanding the above equation
for each coordinate direction gives,
Compatibility Conditions
In the strain-displacement relationships, there
are six strain measures but only three
independent displacements. That is, there are 6
unknowns for only 3 independent variables. As a
result there exist 3 constraint, or compatibility
equations. These compatibility conditions (no
voids created) for infinitesimal strain referred
to rectangular Cartesian coordinates are,
where u, v, and w are the displacements in the x,
y, and z directions respectively (i.e. they are
the components of u).
3D Strain Matrix There are a total of 6 strain
measures. These 6 measures can be organized into
a matrix shown here,
Sect 3.1.2
53
Constitutive relations The Generalized Hooke's
Law
Cauchy generalized Hooke's law to three
dimensional elastic bodies and stated that the 6
components of stress are linearly related to the
6 components of strain. The stress-strain
relationship(s) written in matrix forms, where
the 6 components of stress and strain are
organized into column vectors, are,
e Ss
s Ce
where C is the stiffness matrix, S is the
compliance matrix, and S C-1. In general,
there are 36 stiffness matrix components and the
stress-strain relationships such as these are
known as constitutive relations. However, it
can be shown that conservative materials possess
a strain energy density function and as a result,
the stiffness and compliance matrices are
symmetric. Therefore, only 21 stiffness
components are actually independent in Hooke's
law. The vast majority of engineering materials
are conservative. Please note that the
stiffness matrix is traditionally represented by
the symbol C, while S is reserved for the
compliance matrix. This convention may seem
backwards, but perception is not always reality.
Sect 3.1.2
54
The Problem of Elastic Deformation
Equation of Motion
The Constitutive Relations for Elastic
Materials/Hookes Law
The Strain-Displacement Relations/Including the
Compatibility Equations
A Closed System
Number of Unknowns 15 3 displacements
components, 6 strains and 6 stresses
Number of Equations 15 3 from the equations of
motion, 6 equations from the strain relationships
(including compatibility) and 6 equations from
Hookes Law
Major Problem Materials Properties for the
Constitutive Equations
Sect 3.1.2
55
Generalized Material Properties
Orthotropic Materials
Some engineering materials, including certain
piezoelectric materials (e.g. Rochelle salt) and
2-ply fiber-reinforced composites, are
orthotropic. By definition, an orthotropic
material has at least 2 orthogonal planes of
symmetry, where material properties are
independent of direction within each plane. Such
materials require 9 independent variables (i.e.
elastic constants) in their constitutive matrices.
Anisotropic And Isotropic Materials
In contrast, a material without any planes of
symmetry is fully anisotropic and requires 21
elastic constants, whereas a material with an
infinite number of symmetry planes (i.e. every
plane is a plane of symmetry) is isotropic, and
requires only 2 elastic constants
Sect 3.1.2
56
Hooke's Law in Compliance Form For Orthotropic
Materials
By convention, the 9 elastic constants in
orthotropic constitutive equations are comprised
of 3  Young's modulii Ex, Ey, Ez, the 3 Poisson's
ratios nyz, nzx, nxy, and the 3 shear modulii
Gyz, Gzx, Gxy. The compliance matrix takes the
form,
where
Note that, in orthotropic materials, there is no
interaction between the normal stresses sx, sy,
sz and the shear strains eyz, ezx, exy
Sect 3.1.2
57
Hooke's Law in Stiffness Form For Orthotropic
Materials
The stiffness matrix for orthotropic materials,
found from the inverse of the compliance matrix,
is given by,
where,
The fact that the stiffness matrix is symmetric
requires that the following statements hold,
The factor of 2 multiplying the shear modulii in
the stiffness matrix results from the difference
between shear strain and engineering shear
strain, where
Sect 3.1.2
58
Hooke's Law in Compliance Stiffness Forms For
Isotropic Materials
Hooke's law for isotropic materials in compliance
matrix form is given by,
Some literatures may have a factor 2 multiplying
the shear modulii in the compliance matrix
resulting from the difference between shear
strain and engineering shear strain, where
, etc.
The stiffness matrix is equal to the inverse of
the compliance matrix, and is given by,
Some literatures may have a factor 1/2
multiplying the shear modulii in the stiffness
matrix resulting from the difference between
shear strain and engineering shear strain, where
, etc.
Sect 3.1.2
59
Hooke's Law For Isotropic Materials for Plane
Stresses
For the simplification of plane stress, where the
stresses in the z direction are considered to be
negligible,
the stress-strain compliance relationship for an
isotropic material becomes,
The three zero'd stress entries in the stress
vector indicate that we can ignore their
associated columns in the compliance matrix (i.e.
columns 3, 4, and 5). If we also ignore the rows
associated with the strain components with
z-subscripts, the compliance matrix reduces to a
simple 3x3 matrix,
The stiffness matrix for plane stress is found by
inverting the plane stress compliance matrix, and
is given by,
Note that the stiffness matrix for plane stress
is NOT found by removing columns and rows from
the general isotropic stiffness matrix.
Sect 3.1.2
60
Plane Stress/Isotropic Hooke's Law via
Engineering Strain
Some reference books incorporate the shear
modulus G and the engineering shear strain gxy,
related to the shear strain exy via,
The stress-strain compliance matrix using G and
gxy are,
The stiffness matrix is,
The shear modulus G is related to E and n via,
Poisson's ratio n,
Sect 3.1.2
61
The Thermo-Constitutive Equations in the Elastic
Media
The constitutive equations for a homogeneous,
isotropic body are as follows
The stress-strain compliance relations using G
and gxy are,
The stiffness relations are,
Sect 3.1.2
62
Plane Stresses Strains with Thermal Stress
Consider the previous chart with uniform
increasing or decreasing temperatures
The thermal strain associated with uniform
increasing or decreasing temperatures
The stress-strain compliance relations using G
and gxy are,
The stiffness relations are,
The shear modulus G is related to E and n via,
Sect 3.1.2
63
3.1.2 Thermo-Structural Analysis
3.1.2.1 Thermo-Structural Perspective of Vehicle
Design 3.1.2.2 Review of Strength of
Materials 3.1.2.3 The Mechanics of 1D Linear
Elastic Materials 3.1.2.3.1 Illustrative
Examples of Linear Elastic Materials 3.1.2.4 1D
Linear Thermo-Elastic Problems with
Applications 3.1.2.5 Formulation of the
Thermo-Elastic Problem 3.1.2.5.1 Illustrative
Examples of the Thermo-Elastic Problem
Sect 3.1.2
64
Structural Response in Hypersonic Aircraft Skin
Due to Thermal Loads
Mechanical and thermal buckling behavior of
monolithic and metal-matrix composite
hat-stiffened panels were investigated.
Sect 3.1.2
Ref. NASA TM 4770, Thermal and Mechanical
Buckling Analysis of Hypersonic Aircraft
Hat-Stiffened Panels
65
Problem Aerothermal Loads on Control Surfaces
Sect 3.1.2
66
A Historical Perspective of the YF-12A Thermal
Loads and Structures Program
The YF-12A aircraft with upper surface
temperature contours
This program was one of the most comprehensive
flight and laboratory hot structures and loads
research efforts ever undertaken, Ref. 1. Around
1970, the YF-12A loads and structures efforts
focused on numerous technological issues that
needed defining with regard to aircraft that
incorporate hot structures in the design.
Laboratory structural heating test technology
with infrared systems was largely created during
this program. The program demonstrated the
ability to duplicate the complex flight
temperatures of an advanced supersonic airplane
in a ground-based laboratory, Ref 2.
Ref. 1 NASA YF-12A Flight Loads Program, NASA TM
X-3061, l974. Ref 2 NASA Technical Memorandum
104317, A Historical Perspective of the YF-12A
Thermal Loads and Structures Program, Jerald M.
Jenkins and Robert D. Quinn, 1996
Sect 3.1.2
67
A Historical Perspective of the YF-12A Thermal
Loads and Structures Program
Figure 1
Figure 2
Basic Airplane Structure Figure 1 shows the
structural skeleton of the YF-12A airplane. The
fuselage structure and engine nacelles are formed
as a ring-stiffened structure. The wings are
constructed from multiple spars and ribs. The
aircraft structure was fabricated primarily from
several titanium alloys. Structural Details The
YF-12A airplane designer incorporated several
features within the structure to provide
relief from thermal expansion. Figure 2 shows the
general nature of the wing structure.
Standoff clips were included to minimize the
transference of thermal expansion to the
substructure from the hot skins. These stand-off
clips also minimized the heat conduction paths to
the substructure. The skin panels also included
significant beading to allow flexible thermal
expansion rather than thermal stress build up.
NASA Technical Memorandum 104317, A Historical
Perspective of the YF-12A Thermal Loads and
Structures Program, Jerald M. Jenkins and Robert
D. Quinn
Sect 3.1.2
68
A Historical Perspective of the YF-12A Thermal
Loads and Structures Program
Results of the Mach 3 heating simulation in the
skin, fillet, and spar cap
Time history of typical wing spar temperature
distribution
The general thermal environment of the YF-12A
structure involved a hot skin with a cooler
substructure. In some locations however, the
substructure was hotter than the skin area. These
locations were attributed to either heat from the
engine or hot boundary-layer air entering the
structure through drain holes in the skins. The
heating profile in the above Figure typifies a
Mach 3 flight where an acceleration period
results in increasing skin temperatures. Such
temperatures rapidly come to an equilibrium
situation when the Mach 3 cruise condition is
reached. The substructure lags the skin
temperatures at first, but late in the flight the
substructure reaches near steady state.
NASA Technical Memorandum 104317, A Historical
Perspective of the YF-12A Thermal Loads and
Structures Program, Jerald M. Jenkins and Robert
D. Quinn
Sect 3.1.2
69
A Historical Perspective of the YF-12A Thermal
Loads and Structures Program
Four active arm strain gage bridges were used to
measure flight loads on the YF-12A airplane
Transient Temperature Gradients Leading To
Thermal Stresses And Deformations
NASA Technical Memorandum 104317, A Historical
Perspective of the YF-12A Thermal Loads and
Structures Program, Jerald M. Jenkins and Robert
D. Quinn
Sect 3.1.2
70
The Problem Of Structural Design for Hypersonic
Vehicle

• Structural Strength
• The capacity of the individual elements, which
  together make up the hypersonic structural
  system, to withstand the load (static, dynamic
  transient) that is applied to it.
• Structural Stability
• The capability of a structural system to transmit
  the aero thermal loads safely to the neighboring
  members and eventually to the mainframe.

Sect 3.1.2
71
Sect 3.1.2
72
Sect 3.1.2
73
References

• Advanced Strength of Materials, A. C. Ugural and
  S. K. Fenster
• V. Z. Parton, P. I. Perlin, Mathematical Methods
  of the Theory of Elasticity
• www.efunda.com/
• Thermal Stress, N. Noda, R. B. Hetnarski and Y.
  Tanigawa
• NASA YF-12A Flight Loads Program, NASA TM X-3061,
  l974.
• NASA TM 104317, A Historical Perspective of the
  YF-12A Thermal Loads and Structures Program,
  Jerald M. Jenkins and Robert D. Quinn, 1996
• Thermal Structures for Aerospace Applications,
  Earl Arthur Thornton
• NASA Technical Memorandum 4770, Thermal and
  Mechanical Buckling Analysis of Hypersonic
  Aircraft Hat-Stiffened Panels with Varying Face
  Sheet Geometry and Fiber Orientation, William L.
  Ko

Sect 3.1.2